Metamath Proof Explorer


Theorem mapdpglem3

Description: Lemma for mapdpg . Baer p. 45, line 3: "infer...the existence of a number g in G and of an element z in (Fy)* such that t = gx'-z." (We scope $d g w z ph locally to avoid clashes with later substitutions into ph .) (Contributed by NM, 18-Mar-2015)

Ref Expression
Hypotheses mapdpglem.h 𝐻 = ( LHyp ‘ 𝐾 )
mapdpglem.m 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
mapdpglem.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
mapdpglem.v 𝑉 = ( Base ‘ 𝑈 )
mapdpglem.s = ( -g𝑈 )
mapdpglem.n 𝑁 = ( LSpan ‘ 𝑈 )
mapdpglem.c 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
mapdpglem.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
mapdpglem.x ( 𝜑𝑋𝑉 )
mapdpglem.y ( 𝜑𝑌𝑉 )
mapdpglem1.p = ( LSSum ‘ 𝐶 )
mapdpglem2.j 𝐽 = ( LSpan ‘ 𝐶 )
mapdpglem3.f 𝐹 = ( Base ‘ 𝐶 )
mapdpglem3.te ( 𝜑𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) )
mapdpglem3.a 𝐴 = ( Scalar ‘ 𝑈 )
mapdpglem3.b 𝐵 = ( Base ‘ 𝐴 )
mapdpglem3.t · = ( ·𝑠𝐶 )
mapdpglem3.r 𝑅 = ( -g𝐶 )
mapdpglem3.g ( 𝜑𝐺𝐹 )
mapdpglem3.e ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐺 } ) )
Assertion mapdpglem3 ( 𝜑 → ∃ 𝑔𝐵𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) )

Proof

Step Hyp Ref Expression
1 mapdpglem.h 𝐻 = ( LHyp ‘ 𝐾 )
2 mapdpglem.m 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
3 mapdpglem.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
4 mapdpglem.v 𝑉 = ( Base ‘ 𝑈 )
5 mapdpglem.s = ( -g𝑈 )
6 mapdpglem.n 𝑁 = ( LSpan ‘ 𝑈 )
7 mapdpglem.c 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
8 mapdpglem.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
9 mapdpglem.x ( 𝜑𝑋𝑉 )
10 mapdpglem.y ( 𝜑𝑌𝑉 )
11 mapdpglem1.p = ( LSSum ‘ 𝐶 )
12 mapdpglem2.j 𝐽 = ( LSpan ‘ 𝐶 )
13 mapdpglem3.f 𝐹 = ( Base ‘ 𝐶 )
14 mapdpglem3.te ( 𝜑𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) )
15 mapdpglem3.a 𝐴 = ( Scalar ‘ 𝑈 )
16 mapdpglem3.b 𝐵 = ( Base ‘ 𝐴 )
17 mapdpglem3.t · = ( ·𝑠𝐶 )
18 mapdpglem3.r 𝑅 = ( -g𝐶 )
19 mapdpglem3.g ( 𝜑𝐺𝐹 )
20 mapdpglem3.e ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐺 } ) )
21 20 oveq1d ( 𝜑 → ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) = ( ( 𝐽 ‘ { 𝐺 } ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) )
22 14 21 eleqtrd ( 𝜑𝑡 ∈ ( ( 𝐽 ‘ { 𝐺 } ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) )
23 r19.41v ( ∃ 𝑔𝐵 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ↔ ( ∃ 𝑔𝐵 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) )
24 1 7 8 lcdlmod ( 𝜑𝐶 ∈ LMod )
25 eqid ( Scalar ‘ 𝐶 ) = ( Scalar ‘ 𝐶 )
26 eqid ( Base ‘ ( Scalar ‘ 𝐶 ) ) = ( Base ‘ ( Scalar ‘ 𝐶 ) )
27 25 26 13 17 12 lspsnel ( ( 𝐶 ∈ LMod ∧ 𝐺𝐹 ) → ( 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ↔ ∃ 𝑔 ∈ ( Base ‘ ( Scalar ‘ 𝐶 ) ) 𝑤 = ( 𝑔 · 𝐺 ) ) )
28 24 19 27 syl2anc ( 𝜑 → ( 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ↔ ∃ 𝑔 ∈ ( Base ‘ ( Scalar ‘ 𝐶 ) ) 𝑤 = ( 𝑔 · 𝐺 ) ) )
29 1 3 15 16 7 25 26 8 lcdsbase ( 𝜑 → ( Base ‘ ( Scalar ‘ 𝐶 ) ) = 𝐵 )
30 29 rexeqdv ( 𝜑 → ( ∃ 𝑔 ∈ ( Base ‘ ( Scalar ‘ 𝐶 ) ) 𝑤 = ( 𝑔 · 𝐺 ) ↔ ∃ 𝑔𝐵 𝑤 = ( 𝑔 · 𝐺 ) ) )
31 28 30 bitrd ( 𝜑 → ( 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ↔ ∃ 𝑔𝐵 𝑤 = ( 𝑔 · 𝐺 ) ) )
32 31 anbi1d ( 𝜑 → ( ( 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ↔ ( ∃ 𝑔𝐵 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ) )
33 23 32 bitr4id ( 𝜑 → ( ∃ 𝑔𝐵 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ↔ ( 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ) )
34 33 exbidv ( 𝜑 → ( ∃ 𝑤𝑔𝐵 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ↔ ∃ 𝑤 ( 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ) )
35 df-rex ( ∃ 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ↔ ∃ 𝑤 ( 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) )
36 34 35 bitr4di ( 𝜑 → ( ∃ 𝑤𝑔𝐵 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ↔ ∃ 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) )
37 eqid ( LSubSp ‘ 𝐶 ) = ( LSubSp ‘ 𝐶 )
38 37 lsssssubg ( 𝐶 ∈ LMod → ( LSubSp ‘ 𝐶 ) ⊆ ( SubGrp ‘ 𝐶 ) )
39 24 38 syl ( 𝜑 → ( LSubSp ‘ 𝐶 ) ⊆ ( SubGrp ‘ 𝐶 ) )
40 13 37 12 lspsncl ( ( 𝐶 ∈ LMod ∧ 𝐺𝐹 ) → ( 𝐽 ‘ { 𝐺 } ) ∈ ( LSubSp ‘ 𝐶 ) )
41 24 19 40 syl2anc ( 𝜑 → ( 𝐽 ‘ { 𝐺 } ) ∈ ( LSubSp ‘ 𝐶 ) )
42 39 41 sseldd ( 𝜑 → ( 𝐽 ‘ { 𝐺 } ) ∈ ( SubGrp ‘ 𝐶 ) )
43 eqid ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
44 1 3 8 dvhlmod ( 𝜑𝑈 ∈ LMod )
45 4 43 6 lspsncl ( ( 𝑈 ∈ LMod ∧ 𝑌𝑉 ) → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) )
46 44 10 45 syl2anc ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) )
47 1 2 3 43 7 37 8 46 mapdcl2 ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ∈ ( LSubSp ‘ 𝐶 ) )
48 39 47 sseldd ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ∈ ( SubGrp ‘ 𝐶 ) )
49 18 11 42 48 lsmelvalm ( 𝜑 → ( 𝑡 ∈ ( ( 𝐽 ‘ { 𝐺 } ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ↔ ∃ 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) )
50 36 49 bitr4d ( 𝜑 → ( ∃ 𝑤𝑔𝐵 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ↔ 𝑡 ∈ ( ( 𝐽 ‘ { 𝐺 } ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ) )
51 22 50 mpbird ( 𝜑 → ∃ 𝑤𝑔𝐵 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) )
52 ovex ( 𝑔 · 𝐺 ) ∈ V
53 oveq1 ( 𝑤 = ( 𝑔 · 𝐺 ) → ( 𝑤 𝑅 𝑧 ) = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) )
54 53 eqeq2d ( 𝑤 = ( 𝑔 · 𝐺 ) → ( 𝑡 = ( 𝑤 𝑅 𝑧 ) ↔ 𝑡 = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) ) )
55 54 rexbidv ( 𝑤 = ( 𝑔 · 𝐺 ) → ( ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ↔ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) ) )
56 52 55 ceqsexv ( ∃ 𝑤 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ↔ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) )
57 56 rexbii ( ∃ 𝑔𝐵𝑤 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ↔ ∃ 𝑔𝐵𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) )
58 rexcom4 ( ∃ 𝑔𝐵𝑤 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ↔ ∃ 𝑤𝑔𝐵 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) )
59 57 58 bitr3i ( ∃ 𝑔𝐵𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) ↔ ∃ 𝑤𝑔𝐵 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) )
60 51 59 sylibr ( 𝜑 → ∃ 𝑔𝐵𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) )